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The tame Breuil–Mézard conjecture for unramified groups

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Let GG be an unramified group over Qp\mathbf{Q}_p satisfying Hypothesis, and let d=dim⁡Bˇd=\dim\check{\mathcal{B}} be the dimension of the flag variety of the dual group. For each Serre weight σ\sigma of G(Fp)G(\mathbf{F}_p), let nσ(λ,τ)=[W(λ)⊗σ(τ)‾:σ]n_\sigma(\lambda,\tau)=[\overline{W(\lambda)\otimes\sigma(\tau)}:\sigma]. Tame Breuil–Mézard conjecture. There is a collection of effective cycles Z(σ)∈Zd((XL ⁣GEG⁡)red⁡)\mathcal{Z}(\sigma)\in Z_d((\mathcal{X}_{{}^{L}\!G}^{\operatorname{EG}})_{\operatorname{red}}) such that, for every dominant weight λ\lambda and every tame inertial parameter τ\tau,

[XL ⁣G,Fλ+ρ,τ]=∑σnσ(λ,τ)Z(σ)∈Zd((XL ⁣GEG⁡)red⁡).[\mathcal{X}^{\lambda+\rho,\tau}_{{}^{L}\!G,\mathbf{F}}]=\sum_\sigma n_\sigma(\lambda,\tau)\mathcal{Z}(\sigma)\in Z_d((\mathcal{X}_{{}^{L}\!G}^{\operatorname{EG}})_{\operatorname{red}}).

This is the tame part of the expected Breuil–Mézard conjecture for general unramified groups. The full formulation is currently obstructed by the lack of a sufficiently general inertial local Langlands correspondence.

References

Primary source

Tony Feng and Bao Le Hung, “Mirror symmetry and the Breuil-Mézard Conjecture”, arXiv:2310.07006 (2025).

breuil-mezardp-adic-hodge-theoryrepresentation-theory
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Progress summary

Refreshed
Claimed progress

A 2023 paper proves the predicted cycle formula only for sufficiently generic inputs; the full conjecture remains open.

The conjecture asserts that one collection of effective cycles gives the required cycle identity for every dominant weight and every tame inertial parameter. The full statement is broader than the generic case treated in the available result.

October 2023 generic-parameter result

Mirror symmetry and the Breuil–Mézard Conjecture reports cycles ZEG⁡(σ)\mathcal{Z}^{\operatorname{EG}}(\sigma) satisfying the identity for every λ\lambda and every tame τ\tau that is 2hλ+ρ2h_{\lambda+\rho}-generic, in arbitrary rank. This is claimed progress, not the stated conjecture: nongeneric λ\lambda and τ\tau remain uncovered. A June 2023 paper presents the general tame formulation as conjectural and reports no evidence beyond known special cases.

Current status (as of September 2026): The full conjecture for all dominant λ\lambda and tame τ\tau remains open; only the sufficiently generic case is reported.

Sources

Solutions 0

No solutions have been posted yet.